Bank Marketing

Goal for the project is to Predict whether a client will subscribe (yes / no) to a term deposit, this based on data from a Portuguese bank’s direct marketing campaigns conducted via phone calls.

The dataset is structured and suitable for models like Logistic Regression, LDA, QDA, KNN, Random Forest, and others.

Objectives

  • Build and interpret a logistic regression model without interaction terms.
  • Use EDA and domain intuition (not algorithmic feature selection) to guide variable inclusion.
  • Interpret regression coefficients and confidence intervals, focusing on how key predictor variables influence the likelihood of subscription.
  • Distinguish between statistical significance (p-values, CIs) and practical significance (effect magnitude and meaning).
  • Use AIC as the primary model comparison tool during training, ensuring the model remains interpretable and grounded in insights.

Exploratory Data Analysis

data <- read.csv("./bank-full.csv", header = TRUE, sep = ";", stringsAsFactors = TRUE)

bank <- data |> rename(subscribed = y)
num_rows <- nrow(bank)
num_cols <- ncol(bank) - 1

data_summary <- data.frame(
  Characteristic = c("Number of Rows", "Number of Columns", "Number of Predictors", "Target Variable"),
  Value = c(num_rows, num_cols, num_cols, "subscribed")
)

data_summary

Summary Statistics

summary(bank)
##       age                 job           marital          education    
##  Min.   :18.00   blue-collar:9732   divorced: 5207   primary  : 6851  
##  1st Qu.:33.00   management :9458   married :27214   secondary:23202  
##  Median :39.00   technician :7597   single  :12790   tertiary :13301  
##  Mean   :40.94   admin.     :5171                    unknown  : 1857  
##  3rd Qu.:48.00   services   :4154                                     
##  Max.   :95.00   retired    :2264                                     
##                  (Other)    :6835                                     
##  default        balance       housing      loan            contact     
##  no :44396   Min.   : -8019   no :20081   no :37967   cellular :29285  
##  yes:  815   1st Qu.:    72   yes:25130   yes: 7244   telephone: 2906  
##              Median :   448                           unknown  :13020  
##              Mean   :  1362                                            
##              3rd Qu.:  1428                                            
##              Max.   :102127                                            
##                                                                        
##       day            month          duration         campaign     
##  Min.   : 1.00   may    :13766   Min.   :   0.0   Min.   : 1.000  
##  1st Qu.: 8.00   jul    : 6895   1st Qu.: 103.0   1st Qu.: 1.000  
##  Median :16.00   aug    : 6247   Median : 180.0   Median : 2.000  
##  Mean   :15.81   jun    : 5341   Mean   : 258.2   Mean   : 2.764  
##  3rd Qu.:21.00   nov    : 3970   3rd Qu.: 319.0   3rd Qu.: 3.000  
##  Max.   :31.00   apr    : 2932   Max.   :4918.0   Max.   :63.000  
##                  (Other): 6060                                    
##      pdays          previous           poutcome     subscribed 
##  Min.   : -1.0   Min.   :  0.0000   failure: 4901   no :39922  
##  1st Qu.: -1.0   1st Qu.:  0.0000   other  : 1840   yes: 5289  
##  Median : -1.0   Median :  0.0000   success: 1511              
##  Mean   : 40.2   Mean   :  0.5803   unknown:36959              
##  3rd Qu.: -1.0   3rd Qu.:  0.0000                              
##  Max.   :871.0   Max.   :275.0000                              
## 
str(bank)
## 'data.frame':    45211 obs. of  17 variables:
##  $ age       : int  58 44 33 47 33 35 28 42 58 43 ...
##  $ job       : Factor w/ 12 levels "admin.","blue-collar",..: 5 10 3 2 12 5 5 3 6 10 ...
##  $ marital   : Factor w/ 3 levels "divorced","married",..: 2 3 2 2 3 2 3 1 2 3 ...
##  $ education : Factor w/ 4 levels "primary","secondary",..: 3 2 2 4 4 3 3 3 1 2 ...
##  $ default   : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 2 1 1 ...
##  $ balance   : int  2143 29 2 1506 1 231 447 2 121 593 ...
##  $ housing   : Factor w/ 2 levels "no","yes": 2 2 2 2 1 2 2 2 2 2 ...
##  $ loan      : Factor w/ 2 levels "no","yes": 1 1 2 1 1 1 2 1 1 1 ...
##  $ contact   : Factor w/ 3 levels "cellular","telephone",..: 3 3 3 3 3 3 3 3 3 3 ...
##  $ day       : int  5 5 5 5 5 5 5 5 5 5 ...
##  $ month     : Factor w/ 12 levels "apr","aug","dec",..: 9 9 9 9 9 9 9 9 9 9 ...
##  $ duration  : int  261 151 76 92 198 139 217 380 50 55 ...
##  $ campaign  : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ pdays     : int  -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ...
##  $ previous  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ poutcome  : Factor w/ 4 levels "failure","other",..: 4 4 4 4 4 4 4 4 4 4 ...
##  $ subscribed: Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...

Missing analysis

This data-set contains no empty values at first sight.

colSums(is.na(bank))
##        age        job    marital  education    default    balance    housing 
##          0          0          0          0          0          0          0 
##       loan    contact        day      month   duration   campaign      pdays 
##          0          0          0          0          0          0          0 
##   previous   poutcome subscribed 
##          0          0          0

Separating Numerical vs Categorical

numeric_vars <- names(bank)[sapply(bank, is.numeric)]
categorical_vars <- names(bank)[sapply(bank, function(x) is.factor(x) || is.character(x))]

cat("Numeric variables:\n")
## Numeric variables:
print(numeric_vars)
## [1] "age"      "balance"  "day"      "duration" "campaign" "pdays"    "previous"
cat("Categorical variables:\n")
## Categorical variables:
print(categorical_vars)
##  [1] "job"        "marital"    "education"  "default"    "housing"   
##  [6] "loan"       "contact"    "month"      "poutcome"   "subscribed"

Plot of missing data

Visual confirmation for emptyness search, no data is missing in this data set

vis_miss(bank) +
  labs(
    title = "Visualizing Missing Data",
    x = "",
    y = ""
  ) +
  theme(
    plot.title = element_text(size = 8, face = "bold"),
    plot.subtitle = element_text(size = 8),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )

Find some relationships

plt_subscribed <- bank |>
  group_by(subscribed) |>
  summarise(cnt = n()) |>
  mutate(perc = round(cnt / sum(cnt), 4))

plt_prop <- ggplot(plt_subscribed, aes(x = subscribed, y = perc, colour = subscribed)) +
  geom_bar(aes(fill = subscribed), show.legend = FALSE, stat = "identity") +
  ylab("Proportion of Subscribed")

grid.arrange(grobs = list(tableGrob(plt_subscribed), plt_prop), ncol = 1)

categorical_vars_plt <- categorical_vars[categorical_vars != "subscribed"]

plt_categorical <- lapply(seq_along(categorical_vars_plt), function(i) {
  ggplot(bank, aes_string(x = categorical_vars_plt[i], fill = "subscribed")) +
    geom_bar(position = "fill") +
    scale_y_continuous(labels = scales::percent) +
    scale_fill_discrete() +
    labs(title = paste("Subscription Rate by", categorical_vars_plt[i]),
         y = "Proportion", x = NULL) +
    coord_flip() +
    theme(legend.position = if (i == 1) "bottom" else "none")
})

wrap_plots(plt_categorical, ncol = 3, guides = "collect") &  theme(legend.position = "bottom")

plt_num <- lapply(numeric_vars, function(var) {
  ggplot(bank, aes_string(x = "subscribed", y = var, fill = "subscribed")) +
    geom_boxplot(alpha = 0.7) +
    scale_fill_discrete() +
    labs(title = paste("Dist. of", var, "by Subscribed"), y = var, x = NULL)
})

wrap_plots(plt_num, ncol = 3) & theme(legend.position = "none")

Outlier assesment

for_outliers <- bank
for_outliers$subscribed <- ifelse(bank$subscribed == "yes", 1, 0)
model <- lm(subscribed ~ previous, data = for_outliers)
influencePlot(model, id.method = "identify", 
              main = "Influence Plot: Cook's D vs Leverage",
              sub = "Size of circle = Cook's distance")

Plot the most influential

cooks_d <- cooks.distance(model)
N <- 5
top_influential_df <- data.frame(
  index = 1:length(cooks_d),
  cooks_distance = cooks_d,
  previous = bank$previous,
  subscribed = bank$subscribed
) |>
  arrange(desc(cooks_distance)) |>
  slice(1:N)

ggplot(top_influential_df, aes(x = reorder(as.factor(index), -cooks_distance), y = cooks_distance)) +
  geom_col(fill = "steelblue") +
  geom_text(aes(label = round(cooks_distance, 4)), vjust = -0.5, size = 3.5) +
  labs(
    title = paste("Top", N, "Influential Points (Cook's Distance)"),
    x = "Observation Index",
    y = "Cook's Distance"
  )

Observation 29183 has extremely high leverage and Cook’s Distance (~44.3), making it a highly influential point in the model.

Remove Observation

bank <- bank[-29183, ]

Correlation

for_corr <- bank
for_corr$subscribed <- ifelse(bank$subscribed == "yes", 1, 0)

vars_corr <- names(for_corr)[sapply(for_corr, is.numeric)]
corr_df <- for_corr[vars_corr]

cor_matrix <- cor(corr_df, use = "complete.obs")
subscribed_cor <- cor_matrix[, "subscribed", drop = FALSE]
subscribed_cor <- subscribed_cor[order(abs(subscribed_cor[,1]), decreasing = TRUE), , drop = FALSE]

cor_df <- data.frame(
  variable = rownames(subscribed_cor),
  correlation = subscribed_cor[,1]
)

ggplot(cor_df, aes(x = reorder(variable, correlation), y = correlation)) +
  geom_bar(stat = "identity", fill = "steelblue") +
  coord_flip() +
  labs(title = "Correlation with Subscribed", x = "Variable", y = "Correlation")

PCA

for_pca <- bank
for_pca <- bank[sapply(data, is.numeric)]

pca <- prcomp(for_pca, scale. = TRUE)
autoplot(pca, data = bank, colour = 'subscribed', loadings = TRUE, loadings.label = TRUE) +
  labs(title = "PCA")

loadings <- as.data.frame(pca$rotation)
loadings$variable <- rownames(loadings)

loadings$PC1.ABS <- abs(loadings$PC1)
loadings$PC2.ABS <- abs(loadings$PC2)

top_pc1 <- loadings[order(-loadings$PC1.ABS), c("variable", "PC1")][1:7, ]
top_pc2 <- loadings[order(-loadings$PC2.ABS), c("variable", "PC2")][1:7, ]

top_combined <- data.frame(
  PC1_Variable = top_pc1$variable,
  PC1_Loading = round(top_pc1$PC1, 3),
  PC2_Variable = top_pc2$variable,
  PC2_Loading = round(top_pc2$PC2, 3)
)

top_combined

Principal Component Analysis (PCA) on the numeric variables revealed that pdays and previous contributed most to the first principal component (PC1), capturing variability related to past campaign exposure. The second component (PC2) was primarily influenced by campaign, day, and negatively by duration, reflecting variation in campaign intensity and contact timing.

We can interpret the components as:

  • PC1: Differences in past contact history, primarily driven by pdays and previous.
  • PC2: Differences in campaign intensity and call timing, shaped by campaign, day, and negatively by duration.
eigenvals <- pca$sdev^2
plot(eigenvals / sum(eigenvals), type = "l", main = "Scree Plot", ylab = "Prop. Var. Explained", xlab = "PC #", ylim = c(0, 1))
cumulative.prop <- cumsum(eigenvals / sum(eigenvals))
lines(cumulative.prop, lty = 2)

eigenvals <- pca$sdev^2
prop_var <- eigenvals / sum(eigenvals)
cum_var <- cumsum(prop_var)
pc_table <- data.frame(
  PC = paste0("PC", 1:length(prop_var)),
  "Variance Explained" = round(prop_var, 4),
  "Cumulative Variance" = round(cum_var, 4)
)

pc_table
pca_scores <- as.data.frame(pca$x)
pca_scores$subscribed <- bank$subscribed

plot_ly(
  data = pca_scores,
  x = ~PC1, y = ~PC2, z = ~PC3,
  color = ~subscribed,
  colors = c("red", "deepskyblue"),
  type = "scatter3d",
  mode = "markers"
)

It appears that we’re missing a significant portion of the variance by focusing only on the numeric variables. PC1 and PC2 explain the most variation among these, but even after adding PC3, we don’t observe meaningful separation between subscription outcomes. This suggests that additional structure — possibly critical for understanding or predicting subscribed — may lie in the categorical variables, which were not included in this PCA.

Logistic Regression

Run GLM in each predictor

Let’s understand a bit how the numerals contribute to explain the subscription

for_glm <- bank
for_glm$subscribed <- ifelse(for_glm$subscribed == "yes", 1, 0)

all_num_additive <- glm(subscribed ~ duration + pdays + previous + campaign + balance + age + day, data = for_glm, family = binomial)
summary(all_num_additive)
## 
## Call:
## glm(formula = subscribed ~ duration + pdays + previous + campaign + 
##     balance + age + day, family = binomial, data = for_glm)
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.472e+00  7.700e-02 -45.085  < 2e-16 ***
## duration     3.643e-03  5.648e-05  64.500  < 2e-16 ***
## pdays        1.964e-03  1.552e-04  12.654  < 2e-16 ***
## previous     1.006e-01  7.340e-03  13.709  < 2e-16 ***
## campaign    -1.292e-01  9.609e-03 -13.445  < 2e-16 ***
## balance      3.704e-05  4.294e-06   8.625  < 2e-16 ***
## age          7.910e-03  1.473e-03   5.370 7.88e-08 ***
## day         -1.661e-03  2.014e-03  -0.825    0.409    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 32631  on 45209  degrees of freedom
## Residual deviance: 26464  on 45202  degrees of freedom
## AIC: 26480
## 
## Number of Fisher Scoring iterations: 6

Plot Logistic on Numericals

tidy(all_num_additive, conf.int = TRUE) |>
  filter(term != "(Intercept)") |>
  ggplot(aes(x = reorder(term, estimate), y = estimate)) +
  geom_point() +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.1) +
  coord_flip() +
  labs(title = "Logistic Regression Coefficients",
       y = "Estimate (log-odds)", x = "Variable")

Look for signals in categorical

Modeling with all categoricals might tell some story

bank_cat <- bank
cat_model_vars <- setdiff(categorical_vars, "subscribed")
model_glm_cat <- as.formula(paste("subscribed ~", paste(cat_model_vars, collapse = " + ")))

glm_cats <- glm(model_glm_cat, data = bank_cat, family = binomial)
summary(glm_cats)
## 
## Call:
## glm(formula = model_glm_cat, family = binomial, data = bank_cat)
## 
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)    
## (Intercept)        -1.24333    0.10655 -11.669  < 2e-16 ***
## jobblue-collar     -0.12464    0.06490  -1.921 0.054785 .  
## jobentrepreneur    -0.19100    0.11110  -1.719 0.085581 .  
## jobhousemaid       -0.28253    0.11968  -2.361 0.018238 *  
## jobmanagement      -0.04697    0.06561  -0.716 0.474084    
## jobretired          0.46523    0.07788   5.974 2.32e-09 ***
## jobself-employed   -0.09072    0.09911  -0.915 0.360040    
## jobservices        -0.08617    0.07451  -1.156 0.247478    
## jobstudent          0.33069    0.09770   3.385 0.000713 ***
## jobtechnician      -0.06466    0.06187  -1.045 0.295967    
## jobunemployed       0.13108    0.09768   1.342 0.179626    
## jobunknown         -0.19401    0.20780  -0.934 0.350479    
## maritalmarried     -0.20629    0.05158  -4.000 6.35e-05 ***
## maritalsingle       0.08241    0.05554   1.484 0.137862    
## educationsecondary  0.15081    0.05652   2.668 0.007627 ** 
## educationtertiary   0.31779    0.06570   4.837 1.32e-06 ***
## educationunknown    0.20162    0.09242   2.182 0.029137 *  
## defaultyes         -0.15693    0.14688  -1.068 0.285318    
## housingyes         -0.54527    0.03810 -14.313  < 2e-16 ***
## loanyes            -0.40804    0.05303  -7.694 1.43e-14 ***
## contacttelephone   -0.28127    0.06399  -4.395 1.11e-05 ***
## contactunknown     -1.34752    0.06337 -21.263  < 2e-16 ***
## monthaug           -0.97784    0.06846 -14.284  < 2e-16 ***
## monthdec            0.57038    0.16198   3.521 0.000429 ***
## monthfeb           -0.44644    0.07501  -5.952 2.65e-09 ***
## monthjan           -1.08345    0.10618 -10.203  < 2e-16 ***
## monthjul           -0.79694    0.06766 -11.779  < 2e-16 ***
## monthjun            0.10830    0.08091   1.338 0.180741    
## monthmar            1.06567    0.11027   9.664  < 2e-16 ***
## monthmay           -0.50694    0.06341  -7.995 1.30e-15 ***
## monthnov           -0.83485    0.07443 -11.216  < 2e-16 ***
## monthoct            0.68172    0.09776   6.973 3.10e-12 ***
## monthsep            0.65425    0.10739   6.092 1.11e-09 ***
## poutcomeother       0.25429    0.07960   3.194 0.001401 ** 
## poutcomesuccess     2.26565    0.07345  30.848  < 2e-16 ***
## poutcomeunknown     0.03496    0.05155   0.678 0.497693    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 32631  on 45209  degrees of freedom
## Residual deviance: 27296  on 45174  degrees of freedom
## AIC: 27368
## 
## Number of Fisher Scoring iterations: 6
Reference levels
X_cat <- model.matrix(model_glm_cat, data = bank_cat)

reference_lvls <- data.frame(
  Variable = cat_model_vars,
  Reference = sapply(bank_cat[cat_model_vars], function(x) levels(x)[1])
) |> tibble::as_tibble()


reference_lvls
Odds Ratio
odds_ratios <- data.frame(
  Variable = names(coef(glm_cats)),
  Odds_Ratio = exp(coef(glm_cats))
) |> dplyr::mutate(Effect = paste0(round((Odds_Ratio - 1) * 100, 1), "%")) |>
  dplyr::mutate(Odds_Ratio = round(Odds_Ratio, 3)) |>
  dplyr::arrange(desc(Odds_Ratio)) |>
  tibble::as_tibble()

odds_ratios

We can see from the categorical only variables that:

  • Month of contact has a strong influence: campaigns conducted in March, October, and September significantly increased subscription odds, while those in January and August showed markedly lower performance.
  • Job type and education level also shape likelihood: customers who are retired or students, and those with tertiary education, are notably more likely to subscribe.
  • Additionally, having an existing loan appears to negatively impact subscription odds, suggesting financial burden may reduce campaign receptiveness.
Variable Visual Pattern? Clear % Difference? Keep?
contact yes yes (cellular > unknown) yes
loan yes yes (loan = less likely) yes
housing yes yes (housing = likely) yes
default maybe some maybe
education yes yes (tertiary increases) yes
poutcome strong yes (success = very high) yes
marital maybe some separation maybe
job mixed a few clear signals maybe (group rare levels)

Carefully look at poutcome as we do not know what drives from previous mkt approach, and if the customer is showing an affinity to long term deposit, maybe is increasing the current deposit. Default sounds like a good story, I tried swapping ref with not much difference.

Feature Selection (By EDA)

We are having very conflicting results based on the multiple explorations on numerical, that is telling us, that we need categorical variables to play a role in the explainability.

We think that cyclical encoding for month as we see some patterns on specific months could help the model to explain better as seasonality seems to have some effect.

Variable Type Reason for Inclusion
duration Numerical Strongest univariate predictor; higher durations consistently increase subscription odds
pdays Numerical Captures time since last contact; reflects engagement recency
previous Numerical Reflects past campaign success; useful but may be redundant with pdays/campaign
balance Numerical Indicates client financial status; moderate predictive signal
campaign Numerical Current campaign intensity; negative association suggests fatigue with repeated contact
month_sin Numerical Cyclical encoding of month (seasonality); preserves circular month structure
month_cos Numerical Complement to month_sin; together capture monthly cyclic patterns
contact Categorical Clear visual and statistical difference; contact method affects likelihood to subscribe
loan Categorical Customers with loans are less likely to subscribe; simple and interpretable
education Categorical Higher education levels (tertiary) correlate with higher subscription odds
marital Categorical Some variation observed; potentially useful with clear reference level
job Categorical Certain job roles (retired, student) show increased subscription; use with level grouping

Prepare model

Will do the cyclical encoding for month and dummy variables (as they are factors GLM will dummy them)

candidate_data <- bank
candidate_data$month_num <- as.numeric(factor(candidate_data$month, levels = c(
  "jan", "feb", "mar", "apr", "may", "jun",
  "jul", "aug", "sep", "oct", "nov", "dec"
)))

candidate_data$month_sin <- sin(2 * pi * candidate_data$month_num / 12)
candidate_data$month_cos <- cos(2 * pi * candidate_data$month_num / 12)


candidate_data <- candidate_data |> dplyr::select(-month)
head(candidate_data)
split_rate <- 0.7
split <- sample(1:nrow(candidate_data), split_rate * nrow(candidate_data))
train_data <- candidate_data[split, ]
test_data  <- candidate_data[-split, ]

#num_feat <- c("duration", "poutcome", "pdays", "balance", "default", "housing", "campaign", "month_sin", "month_cos")
num_feat <- c("duration", "poutcome", "balance", "housing", "campaign", "month_sin", "month_cos")
cat_feat <- c("contact", "loan", "education", "marital", "job", "day", "previous")

features <- c(num_feat, cat_feat)
candidate_model <- as.formula(paste("subscribed ~", paste(features, collapse = " + ")))

GLM

threshold <- 0.25
candidate_fit <- glm(candidate_model, data = train_data, family = binomial)
glm_pred <- predict(candidate_fit, newdata = test_data, type = "response")

model_levels <- levels(candidate_data$subscribed)
pred_class <- factor(ifelse(glm_pred > threshold, "yes", "no"), levels = model_levels)
glm_actual <- factor(test_data$subscribed, levels = model_levels)

confusionMatrix(pred_class, glm_actual, positive = "yes")
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    no   yes
##        no  11261   717
##        yes   734   852
##                                           
##                Accuracy : 0.893           
##                  95% CI : (0.8877, 0.8982)
##     No Information Rate : 0.8843          
##     P-Value [Acc > NIR] : 0.0007205       
##                                           
##                   Kappa : 0.4796          
##                                           
##  Mcnemar's Test P-Value : 0.6744593       
##                                           
##             Sensitivity : 0.54302         
##             Specificity : 0.93881         
##          Pos Pred Value : 0.53720         
##          Neg Pred Value : 0.94014         
##              Prevalence : 0.11567         
##          Detection Rate : 0.06281         
##    Detection Prevalence : 0.11693         
##       Balanced Accuracy : 0.74091         
##                                           
##        'Positive' Class : yes             
##

Tune thresholds

thresholds <- seq(0.1, 0.9, by = 0.01)

metrics_df <- purrr::map_dfr(thresholds, function(thresh) {
  pred_class <- factor(ifelse(glm_pred > thresh, "yes", "no"), levels = c("no", "yes"))
  
  tibble(
    threshold = thresh,
    precision = yardstick::precision_vec(truth = glm_actual, estimate = pred_class, event_level = "second"),
    recall = yardstick::recall_vec(truth = glm_actual, estimate = pred_class, event_level = "second"),
    f1 = yardstick::f_meas_vec(truth = glm_actual, estimate = pred_class, event_level = "second")
  )
})

ggplot(metrics_df, aes(x = threshold)) +
  geom_line(aes(y = f1), color = "blue") +
  geom_line(aes(y = precision), color = "green") +
  geom_line(aes(y = recall), color = "red") +
  labs(title = "Threshold Tuning", y = "Metric", x = "Threshold")

Misclassification spot-check

test_data_aug <- test_data %>%
  dplyr::mutate(
    pred_prob = glm_pred,
    pred_class = factor(ifelse(glm_pred > threshold, "yes", "no"), levels = c("no", "yes")),
    subscribed = factor(subscribed, levels = c("no", "yes")),
    result = case_when(
      subscribed == "yes" & pred_class == "yes" ~ "TP",
      subscribed == "no" & pred_class == "yes" ~ "FP",
      subscribed == "yes" & pred_class == "no" ~ "FN",
      subscribed == "no" & pred_class == "no" ~ "TN"
    )
  )

ggplot(test_data_aug, aes(x = duration, y = balance, color = result)) +
  geom_point(alpha = 0.4) +
  labs(title = "False Positives vs. True Positives",
       subtitle = paste("Threshold:", threshold),
       color = "Prediction Outcome")

test_data$subscribed <- factor(test_data$subscribed, levels = c("no", "yes"))

pr_df <- tibble(
  subscribed = test_data$subscribed,
  .pred_yes = glm_pred
)

pr_curve(pr_df, truth = subscribed, .pred_yes) %>%
  autoplot() +
  labs(
    title = "Precision-Recall Curve",
    subtitle = "Probability thresholds for predicting 'yes'"
  )

Prediction - Objective 2

#' Local Variables
#' -  Setting threshold for all models - might need adjustment for each; but the goal is to compare in a fair ground
#' -  Train Control for all models, same cross validation. Can be set individually for each model
#' -  Set Reference Levels

threshold <- 0.25
model_levels <- levels(candidate_data$subscribed)

fitControl <- trainControl(
  method = "repeatedcv",
  number = 10,
  repeats = 1,
  classProbs = TRUE,
  summaryFunction = mnLogLoss
)

Quick inspection for potential complexity

bank_loess <- bank
bank_loess$subscribed <- ifelse(bank_loess$subscribed == "yes", 1, 0)
plt_numeric_vars <- setdiff(numeric_vars, c("day"))

plots <- lapply(plt_numeric_vars, function(var) {
  ggplot(bank_loess, aes_string(x = var, y = "subscribed", colour = var)) +
    geom_point() +
    geom_smooth(method = "loess", se = FALSE, size = 1, span = 1.5) +
    ylim(-.2, 1.2) +
    labs(title = paste("Subscription vs", var), y = "Subscription Rate", x = var)
})
grid.arrange(grobs = plots, ncol = 3)

Variable Shape Interpretation Suggestion
age U-shape → increasing Older clients tend to subscribe more; age shows a non-linear effect Include age^2
balance Inverted U-shape Clients with mid-range balances are more likely to subscribe Use balance + balance^2
duration Logistic-like rise then plateau Longer calls are associated with higher subscription rates, with signs of saturation Consider log(duration + 1) or natural spline
campaign Shallow U-shape Additional contacts may reduce effectiveness after a point Keep as linear or bin into tiers
pdays Flat with mild positive curvature Values like -1 dominate; higher values show a weak increasing trend Cap at 300 or bin; optionally include pdays == -1 indicator
previous Curved rise then decline Moderate prior contact increases success; too many may reduce effectiveness Use previous + previous^2 or log(previous + 1)

Complex Logistic Regresion (GLM)

set.seed(42)
glm_fit <- train(
  candidate_model,
  data = train_data,
  method = "glm",
  family = "binomial",
  trControl = fitControl,
  metric = "logLoss"
)

glm_probs <- predict(glm_fit, newdata = test_data, type = "prob")[, "yes"]
glm_preds <- factor(ifelse(glm_probs > threshold, "yes", "no"), levels = model_levels)
glm_actual <- factor(test_data$subscribed, levels = model_levels)

confusionMatrix(data = glm_preds, reference = glm_actual, positive = "yes")
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    no   yes
##        no  11261   717
##        yes   734   852
##                                           
##                Accuracy : 0.893           
##                  95% CI : (0.8877, 0.8982)
##     No Information Rate : 0.8843          
##     P-Value [Acc > NIR] : 0.0007205       
##                                           
##                   Kappa : 0.4796          
##                                           
##  Mcnemar's Test P-Value : 0.6744593       
##                                           
##             Sensitivity : 0.54302         
##             Specificity : 0.93881         
##          Pos Pred Value : 0.53720         
##          Neg Pred Value : 0.94014         
##              Prevalence : 0.11567         
##          Detection Rate : 0.06281         
##    Detection Prevalence : 0.11693         
##       Balanced Accuracy : 0.74091         
##                                           
##        'Positive' Class : yes             
##

LDA Model

set.seed(42)
lda_fit <- train(
  candidate_model,
  data = train_data,
  method = "lda",
  trControl = fitControl,
  metric = "logLoss"
)

lda_actual <- factor(test_data$subscribed, levels = model_levels)
lda_probs <- predict(lda_fit, newdata = test_data, type = "prob")[, "yes"]
lda_preds <- factor(ifelse(lda_probs > threshold, "yes", "no"), levels = model_levels)

confusionMatrix(data = lda_preds, reference = lda_actual, positive = "yes")
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    no   yes
##        no  11345   778
##        yes   650   791
##                                           
##                Accuracy : 0.8947          
##                  95% CI : (0.8894, 0.8998)
##     No Information Rate : 0.8843          
##     P-Value [Acc > NIR] : 6.691e-05       
##                                           
##                   Kappa : 0.4665          
##                                           
##  Mcnemar's Test P-Value : 0.0007772       
##                                           
##             Sensitivity : 0.50414         
##             Specificity : 0.94581         
##          Pos Pred Value : 0.54892         
##          Neg Pred Value : 0.93582         
##              Prevalence : 0.11567         
##          Detection Rate : 0.05832         
##    Detection Prevalence : 0.10624         
##       Balanced Accuracy : 0.72498         
##                                           
##        'Positive' Class : yes             
##

Random Forest

set.seed(42)
rf_fit <- train(
  candidate_model,
  data = train_data,
  method = "ranger",
  trControl = fitControl,
  metric = "logLoss",
  tuneLength = 2
)

rf_probs <- predict(rf_fit, newdata = test_data, type = "prob")[, "yes"]
rf_actual <- factor(test_data$subscribed, levels = model_levels)
rf_preds <- factor(ifelse(rf_probs > threshold, "yes", "no"), levels = model_levels)

ROC Metrics

roc_lda <- roc(lda_actual, lda_probs, levels = model_levels, direction = "<")
roc_glm <- roc(glm_actual, glm_pred, levels = model_levels, direction = "<")
roc_rf <- roc(rf_actual, rf_probs, levels = model_levels, direction = "<")

plot(roc_lda, col = "blue", lwd = 2, main = "ROC Curve: LDA vs Logistic vs Random Forest")
lines(roc_glm, col = "firebrick", lwd = 2)
lines(roc_rf, col = "green", lwd = 2)

legend("bottomright",
  legend = c(
    paste("LDA           (AUC =", round(auc(roc_lda), 3), ")"),
    paste("Logistic      (AUC =", round(auc(roc_glm), 3), ")"),
    paste("Random Forest (AUC =", round(auc(roc_rf), 3), ")")
  ),
  col = c("blue", "firebrick", "green"),
  lwd = 2
)

Performance Metrics

cm_lda <- confusionMatrix(lda_preds, lda_actual, positive = "yes")
cm_glm <- confusionMatrix(pred_class, glm_actual, positive = "yes")
cm_rf <- confusionMatrix(rf_preds, rf_actual, positive = "yes")


custom.logloss <- function(probs, actual, eps = 1e-15) {
  probs <- pmin(pmax(probs, eps), 1 - eps)
  actual_binary <- ifelse(actual == "yes", 1, 0)
  -mean(actual_binary * log(probs) + (1 - actual_binary) * log(1 - probs))
}

custom.metrics <- function(cm, roc, probs, actual) {
  sensitivity <- cm$byClass["Sensitivity"]
  ppv <- cm$byClass["Pos Pred Value"]
  f1_score <- 2 * ((sensitivity * ppv) / (sensitivity + ppv))

  data.frame(
    Accuracy    = cm$overall["Accuracy"],
    Sensitivity = sensitivity,
    Specificity = cm$byClass["Specificity"],
    PPV         = ppv,
    NPV         = cm$byClass["Neg Pred Value"],
    Prevalence  = cm$byClass["Prevalence"],
    AUROC       = auc(roc),
    F1          = f1_score,
    LogLoss     = custom.logloss(probs, actual)
  )
}

results_table <- rbind(
  LDA = custom.metrics(cm_lda, roc_lda, lda_probs, lda_actual),
  Logistic = custom.metrics(cm_glm, roc_glm, glm_probs, glm_actual),
  RandomForest = custom.metrics(cm_rf, roc_rf, rf_probs, rf_actual)
)

round(results_table, 4)